To analyze the problem, let's examine each limit based on the graph of the function $f(x)$ provided:

1. $\lim_{x \to 1^+} f(x)$:

   • As $x$ approaches 1 from the right (positive side), we observe that $f(x)$ approaches the y-value at the solid dot located at $(1, 1)$.
   • Therefore, $\lim_{x \to 1^+} f(x) = 1$.

2. $\lim_{x \to 1^-} f(x)$:

   • As $x$ approaches 1 from the left (negative side), $f(x)$ approaches the y-value where there is an open circle at $(1, -1)$.
   • Thus, $\lim_{x \to 1^-} f(x) = -1$.

3. $\lim_{x \to 1} f(x)$:

   • Since the left-hand limit and right-hand limit at $x = 1$ do not match ($-1 \neq 1$), the overall limit does not exist.
   • Therefore, $\lim_{x \to 1} f(x)$ does not exist.

4. $\lim_{x \to 2} f(x)$:

   • As $x$ approaches 2, we observe that the function has a jump discontinuity, with the open circle at $(2, 1)$ and a solid dot at $(2, -1)$.
   • Thus, the limit does not exist because there is no single value that $f(x)$ approaches as $x \to 2$.
   • So, $\lim_{x \to 2} f(x)$ does not exist.

5. $\lim_{x \to 3} f(x)$:

   • As $x$ approaches 3, the function appears to approach a solid dot at $(3, -1)$.
   • Therefore, $\lim_{x \to 3} f(x) = -1$.

6. $\lim_{x \to -1} f(x)$:

   • As $x$ approaches $-1$, $f(x)$ seems to approach the point $( -1, 0 )$.
   • So, $\lim_{x \to -1} f(x) = 0$.

Would you like further clarification on any of these calculations or have specific questions?

Here are five questions to deepen your understanding:

1. Why is the limit nonexistent at points where there is a jump discontinuity?
2. What is the significance of one-sided limits when evaluating the behavior of a function near discontinuities?
3. How would the presence of a closed or open circle affect the continuity of a function at a given point?
4. Could you determine if this function is continuous on any intervals of $x$?
5. What role does the limit play in understanding the behavior of functions around points of interest?

Tip: When evaluating limits graphically, always pay close attention to open and closed circles, as they indicate whether the function actually reaches certain values or merely approaches them.